In a natural plasma environment the two instabilities at play for the generation of LHW are (i) the modified-two-stream instability (hereafter MTSI) driven by a supra-thermal ion beam [1]}, [2]}, [3]} and (ii) the lower-hybrid-drift instability (hereafter LHDI) driven by the relative drift between ions and electrons [4]}, [5]}, [6]}.
The electron acceleration driven by LHW generated by the MTSI has been widely addressed in the literature using quasilinear theory [2]}, [8]}, full-kinetic simulations [9]}, [10]} and experiments [11]}.
The MTSI is considered to be the typical source for the above mentioned LHW observations at plasma shock fronts [12]}, [13]}, [8]} due to the reflection of a large fraction of solar wind ions by the shock front; however, LHW are routinely observed also in the absence of such beams of reflected ions, a condition that prevents to invoke the MTSI as the underlying mechanism.
In such cases, the LHW are instead generated by the LHDI. The driver for the development of the LHDI has to be found in the “strong” density gradients reaching length scales of the order of, or even shorter than, the ion gyro-radius. This is the case, for instance, for the above-mentioned observations at Earth's magnetosphere and in cometary plasmas.
From now on, we shall focus on LHW generated by the LHDI and their interaction with the electron population.
